Fractional Differential Equations in Mathematical Biology – modelling and simulation.

Project description

In this PhD project we will explore modelling applications in mathematical biology based on fractional differential equations. Our goal is to make use of fractional models in order to model spike frequency adaptation of single neurons and to analyse coupled systems in order to identify emergent phenomena that result from the interplay of large ensembles of neurons. Another field of application we will address concerns protein friction in cytoskeleton networks. We intend to model protein-protein interaction resulting in stick-slip behaviour using fractional models. We will also see the effect of variable order fractional differential equations with different kernels in modelling the same phenomena.

To run the simulations, we will develop numerical schemes that reduce the memory footprint of the memory tail characterising fractional derivatives and variable order fractional derivatives. Moreover, Caputo fractional derivative of smooth function may also possess singularity which affects the order of the method. Therefore, high order approximation to Caputo fractional derivative on graded mesh will also be derived. The convergence and stability of the difference scheme will also be proved. The involvement of an additional parameter (order of the derivative) demands more efficiency of numerical schemes as optimal parameter can be learned using hit and trial methods. Using machine learning, the parameters will be directly learned from the data in a highly efficient manner without solving the inverse problem.

Outcomes

Efficient numerical schemes with little memory footprint which allow to identify pseudo-stationary states of fractional differential equation/variable order fractional differential equation models. Detailed analysis of the potential of fractional models to capture spike-frequency adaptation of neuron cells. Formulation and simulation of the collective activity of neurons based on fractional models. Mathematical models for protein friction and simulation codes for cytoskeleton networks. Framework that can simultaneously solve a system of variable order FDEs and estimate its parameters.

Essential capabilities

Mathematics, Programming.

Desireable capabilities

Fundamentals of Physics and Biology.

Expected qualifications (Course/Degrees etc.)

Mathematics, Physics, Engineering (with mathematics orientation).

Candidate Discipline

Fractional Equation Neuron spiking Modelling Numerical methods.